In this explainer, we will learn how to relate volumes of different shapes, surface areas of different shapes, and a shape’s volume and surface area.
Since we are interested in relating the volumes and surface areas of shapes, let us begin by recalling the formulas that exist for calculating these quantities. First of all, let us consider probably the simplest case, spheres.
Formula: Volumes and Surface Areas of Spheres
Let be the radius of a sphere. Then, its volume is given by and the surface area is given by
Note that both formulas only have one variable quantity: the radius . This means that, if we are given the radius, we can find both the volume and the surface area. However, that is not all. Suppose we are given the surface area of a sphere and want to find other measurements of the sphere. Since the surface area is we can find the radius by rearranging the formula in terms of it. We first divide both sides by to get
Then, as is always positive, we can take the square root of both sides:
So, from the surface area, we can get the radius. Furthermore, since the volume is we can substitute directly into this equation to find the volume as well.
This process also applies the other way around. If we are given the volume, we can find the radius and, by extension, the surface area. Let us see an example of this.
Example 1: Relating the Volume and Surface Area of a Sphere
Given that the volume of a sphere is cm3, find its surface area in terms of .
We begin by considering the information given to us and what relevant formulas we can apply. We have been given the volume of a sphere, which we can recall is given by where is the radius of the sphere. We have also been asked to find the surface area , which is given by
Since both formulas involve , we can relate the two. Specifically, we can use the volume formula to find the radius, which we can then substitute into the surface area formula to find . We start by substituting into the volume formula to get
Dividing both sides by , we have
Then, to find , we can take the cube root of both sides. This gives us
Now, we can substitute into the surface area formula. This gives us
Since we have been asked to find the surface area in terms of , we can leave the answer as it is without simplifying.
As we saw in the last example, the volume and surface area of a sphere are closely related to each other. This also applies to other kinds of shapes; for instance, let us consider cylinders. First, let us recall the formulas for the volume and surface area of a cylinder.
Formula: Volumes and Surface Areas of Cylinders
For a cylinder, let be the radius of the base and let be the height. Then, its volume is given by and its surface area is given by
Recall that both of these formulas come from the formulas for the area and circumference of a circle. Firstly, the volume is the area of the circular base times the height , which is shown in the left-hand diagram below. For the surface area, we consider the net of the cylinder, which is the area of two circles , plus the curved area, which is the circumference times the height . This is shown in the right-hand diagram below.
Considering the above formulas, we see that they are both in terms of two variables: the radius and the height . We highlight this similarity below.
In other words, as long as we have the radius and height, it is possible to find both the volume and surface area. Additionally, if we are given either the radius or the height and either the volume or the surface area, it is possible to find the other two measurements. This is because we can rearrange the formulas to be in terms of the missing variables.
To show how this works, let us consider an example where we are given the curved surface area and the diameter of a cylinder and need to find the volume.
Example 2: Relating the Volume and Surface Area of a Cylinder
A cylinder has a curved surface area of 656 cm2 and a diameter of 10.2 cm. Find its volume, giving your answer to the nearest cubic centimetre.
Let us first of all understand the information given to us. We have been given that the curved surface area is 656 cm2, which is a portion of the total surface area, as shown below.
In other words, the curved surface area is the area of the rectangle shown, which is times . Additionally, we have been given that the diameter is 10.2 cm, where the diameter is twice the radius . Since we have been given two measurements of the cylinder, we can use them to find the height, and consequently, the volume.
First, for the curved surface area, we have
Using the fact that , where is the diameter, we have
To find , we divide both sides by to get
Now, we can use this to find the volume. Recall that the volume of a cylinder is
We substitute in and to get
One other type of shape we will commonly see is cuboids (and cubes). Let us recall the formulas for the volume and surface area of a cuboid.
Formula: Volume and Surface Area of Cuboid
Suppose a cuboid has length , width , and height , as shown in the figure below. Then, its volume is given by and its surface area is given by
The formula for the volume is fairly straightforward, since it is just the product of the three dimensions of the cuboid. For the surface area, recall that this can be found by considering the net of the cuboid, as shown below.
In other words, the areas of each of the 6 faces are added together, where the area of each face is the product of two dimensions of the cuboid.
We also note that the above formulas easily extend to cubes, since a cube is just a cuboid with equal sides. In this case, , so the volume becomes and the surface area becomes
So far, we have seen examples of relating the surface area to the volume of individual shapes, but it is equally possible to relate the volumes and surface areas of different shapes together. Let us consider a case where a sphere is contained within a cube.
Example 3: Relating the Volumes of a Cube and a Sphere
If a sphere is inscribed in a cube of volume 8 cm3, what is the volume of the sphere?
The key to this question is being able to relate the dimensions of the cube to the sphere. We note that the sphere is inscribed in the cube, which means that the sphere is touching each face of the cube without any gaps. This means that the diameter of the sphere is equal to the length of the cube (or alternatively, its radius is half that of the length of the cube). This is demonstrated below.
Therefore, if we can find the length of the cube, then we can find the diameter (and radius) of the sphere.
Now, we have been given that the volume of the cube is 8 cm3. Recall that if the length of a cube is , then its volume is given by
Substituting , we have
We can obtain by taking the cube root of both sides of the equation. This gives us
As mentioned earlier, the radius is half this length. So, this means that
Now that we have the radius of the sphere, we can obtain its volume by using the formula
Since , we have
As demonstrated in the previous example, when comparing different objects together, it is often important to recognize when their dimensions such as the length, height, or radius are related to each other. We can do this using information given to us in the problem and deductive reasoning. Let us see another example of relating two objects together.
Example 4: Relating the Surface Areas of a Cuboid and a Cylinder
If the total surface of a rectangular prism is 40 cm2 and its height is twice its width, find the total surface area of a cylinder inscribed inside the rectangular prism.
Let us begin by analyzing the information given in the question. The cylinder has been inscribed within a cuboid, which means that the edges of the two shapes are touching. Let us consider what this means by looking at a cross-section (i.e., a top-down view) of the two shapes.
Here, we have labeled the two diameters of the cylinder , the cuboid’s width , and the cuboid’s length . We can see from the diagram that the width and length must each be equal to the diameters, which means .
Secondly, from the question, we have been given that the height is twice the width (i.e., ). This means that all three dimensions of the cuboid can be expressed in terms of the width. We label this on the diagram below.
Here, is the radius of the cylinder, and we note that the height of the cylinder is the same as the height of the cuboid, .
Let us now apply the formulas we know for the surface areas of cylinders and cuboids. Ordinarily, the surface area of a cuboid is given by but here, we have and . We have also been given that the total surface area is 40 cm2. Substituting these values, we get
From this, we can find the width. We first divide by 10 on both sides to get and, since is a length (i.e., it must always be positive) we can take the square root of both sides as follows:
Now, recall that the radius of the cylinder is half of , so . We now recall that the total surface area of a cylinder is the area of the top and bottom circles plus the circular portion, which is
Here, we have cm and , so we substitute these values into the formula to get
For our final example, we will consider a situation where multiple spheres are combined together and we have to consider their relation to a cylinder.
Example 5: Relating the Volumes and Surface Areas of Multiple Shapes
Three spheres are inscribed in a cylinder, as shown in the figure. If the volume of a sphere is cm3, find the total surface area of the cylinder.
In this question, we need to consider how a single sphere relates to the dimensions of the cylinder that contains the three spheres. To do this, we can make use of the fact that the spheres are inscribed, which means they fit exactly into the cylinder with no gaps in the top, bottom, or sides. In particular, this means the radii of the spheres and the cylinder must be the same. It also means that the height of the cylinder must be equal to 3 times the diameter of a single sphere (or 6 times the radius of one). Letting be the radius, this is illustrated below.
To find the surface area of the cylinder, it will be necessary to find since that will give us the radius and height of the cylinder.
Since we have been given the volume of a sphere, let us recall that the volume is related to by the formula
Letting , we have
To get from here, we can rearrange the equation by dividing both sides by . This gives us
Next, we can find by taking the cube root of both sides. This gives us
Now, let us recall the formula for the surface area of a cylinder. This is equal to the area of the two circles forming the top and bottom plus the area of the curved region, which is equal to
We have already shown that for the cylinder since it has the same radius as one of the spheres, and the height is . Substituting these values in, we get
Let us finish by considering the main things we have learned in this explainer.
- For a sphere, if we have any one of the volume , radius , or surface area , we can find the other two measurements using the formulas
- For a cylinder, if we have the radius and height , then we can find the volume and surface area using the formulas Furthermore, if we are given either the volume or the surface area and either the radius or the height, we can calculate the other two measurements by rearrangement.
- For a cuboid (or cube), if we have the length , width , and height , then we can find the volume and surface area using the formulas Additionally, if we are given two of the dimensions of a cuboid and either the volume or surface area, we can find the remaining measurements by rearrangement.
- In problems where we are given different shapes, we can relate them to each other by using the information given to us in the problem and the rearrangement of the above formulas.